AMC 10 · 2004 · #2

Grade 6 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
exponentsorder-of-operations extremal-constructionsystematic-enumeration ↑ Prerequisites: exponents
📏 Medium solution 💡 2 insights
Problem
The four numbers 0, 1, 2, and 3 are placed, one each, into a, b, c, and d in the expression c times a to the b, minus d. Find the largest value the expression can reach.

Pick an answer.

(A)
5
(B)
6
(C)
8
(D)
9
(E)
10
How to solve
Strategy Extreme Principle

The question asks for a maximum, the signature trigger for Tool #14 (Extreme Principle): push each part of the expression toward the extreme that makes the whole biggest — subtract the smallest number you can, and build the largest product you can. Tool #2 (Make a Systematic List) keeps the few strong candidates organized so none is missed, and Tool #3 (Eliminate Possibilities) rules out the traps — wasting a big number as the multiplier, or settling for the second-best power — to confirm 9 is truly the ceiling.

1STEP 1

Subtract the least, build the rest

Spend the 0 on the subtraction, where it takes nothing away.

d=0 → c · a^b-d=c · a^b
2STEP 2

The power grows fastest

The power climbs fastest, and 3 squared beats 2 cubed.

3²=9 ＞ 2³=8
3STEP 3

Assign the leftovers

The leftover 1 must multiply, since a 0 there would wipe out everything.

c · a^b-d=1 · 3²-0=1 · 9-0=9
4STEP 4

Check the other contenders

Every rival arrangement lands at 8 or below, so the maximum is 9, choice (D).

1 · 8-0=8, 3 · 2¹-0=6, 2 · 3¹-0=6 → max=9 (D)
Answer
9
The result cannot exceed the largest product you can build, and that product tops out at 1 · 3²=9; since you then subtract d ≥ 0, the whole expression is at most 9. That instantly rules out 10 (choice E) as unreachable — there is no negative number here to subtract and push the total higher. The trap answers match specific mistakes: 8 comes from settling for the power 2³, and 6 comes from wasting a big number as the multiplier on a small power. Landing exactly on the offered choice 9 is a good sign the assignment is right.
💡Key takeaway

To make an expression as big as possible, feed the fastest-growing part — the exponent — your biggest numbers, never multiply by 0, and save the 0 for whatever you subtract.

  • Subtract the least, build the rest
  • The power grows fastest
  • Assign the leftovers
  • Check the other contenders

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